A more detailed description can be found as a
Danish Maritime Institute:
Dynamic Positioning System.
The problem is how to most efficiently ensure that a ship, floating on
the surface of the ocean, stays very nearly at rest with respect to
the sea bed. One may employ thrusters, propellers and rudder
action. More precisely, given external forces from wind, current and
waves, and a specified position and motion envelope for the ship bulk,
study the number, type, and placement of dynamic motion controls on
the bulk with the goal of forming an optimal Dynamic Positioning
Air-conditioning scroll compressors are manufactured by the million
today and get greater and greater market shares from traditional
(reciprocating) single piston compressors.
The scroll compressor consist of two plane spiral/helix running inside each other. Normally both scrolls are the same unrolling circle involute with constant wall thickness. The compression chambers are therefore thin, oblong, and bent. Often the one scroll is fixed and the other is orbiting.
The scroll compressor has its suction port at the periphery and discharge in the spiral centre. No suction valve is needed and a discharge valve is only present to decrease power consumption a little bit.
The scroll compressor has a rather constant speed of compression chamber decrease and thus a harmless almost constant torque loading of partly the electric motor and partly the housing (action and reaction).
Consequences of the scroll geometry are long gas leakage passages and high material temperatures in the scroll centre.
Different movements of scrolls are topics of today: Co-rotating and co-orbiting scrolls.
The task could be: Chose one scroll geometry and movements of both
scrolls. Find the other scroll geometry (One scroll is envelope of the
other). Scroll wall thickness could be a variable too. Make
investigations and see how compressor performance/efficiency is
influenced. Finally, find sensibilities and optimise design if possible.
LEGO: How to build with LEGO.
The idea for the project is simple - if any 3D body is given, how can it be built with LEGO bricks?
Unit-volume (the smallest possible volume) in the LEGO universe is a so-called
"generic LEGO brick". It is a brick 8mm long and wide, and 3.2mm high,
and has only one position ("stud") for connecting with other bricks.
Although there are a lot of different LEGO bricks, in this project the use of
"family" bricks only is allowed. "Family" LEGO bricks are paralelopiped-shape
bricks that can be made of "generic" LEGO bricks, by putting several of
them next to each other and/or above each other.
Allowed dimensions of "family" LEGO bricks are: (if the dimensions of the generic LEGO brick are set to 1,1,1)
Please note that some bricks appear in different heights. Length and width of bricks correspond to their number of studs (connection points) in the horizontal plane.
The usage of LEGO DUPLO bricks is allowed also. Dimensions of DUPLO bricks,
in "generic brick" measure, are:
There is one important restriction here: Due to construction, DUPLO bricks can be connected only to family bricks with an even number of generic bricks in the length and width dimensions, and only with the bricks of height at least 3.
So the task is:
For a "legoized" 3D model (i.e. a 3D model that is represented as a set of 1x1x1 generic LEGO bricks put next to each other or above each other), find an algorithm to build the model of actual LEGO bricks from the previous tables, so that model should stand connected. We assume that a brick is connected to a model if at least two of its connection points are connected to a model. (If the brick has less than three connection points, then it is enough that only one of its connection points is connected to a model).
We do not want a model to be solid, meaning "full of bricks". Whenever
it is possible to make an invisible hole inside a model, we would like it
to be done - to spend less bricks in building. But all the bricks should
be connected to the model as described earlier. A good "rule of thumb"
should be that the width of the "wall" from outside to inside of the model
is 4 connection points (it can be more or less in some places, the shape of
the inside hole is not important).
We can assume nothing about the shape of a legoized object except that it is connected. It can have any number of holes into it (a cup with two handles, for example).
The preferred output of this project should be an algorithm for making a computer program for determining which brick should be put in which place. One of the problems we can see now is that there is not a unique solution - i.e. every model can be built using many different bricks. A criterion that "more solid" models are preferred can be used to set some cost-function. "More solid" models are models made with bigger bricks, and with bricks that have more connection points connected (and even these two criterions can be opposite!).
I have tried to explain here a problem as it appears in "real life".
I hope this is enough for you to make an exact mathematical formulation
of the problem (although I am aware that, as with all "real life" problems,
some terms and requirements are not defined quite precisely - we require
that the outside of the final model is exactly as in the legoized model,
but we do not have such precise requests about the insides of the model,
not even the precise thickness of the "wall" - but, these are facts of life!)
Mixing of clorine in swimmingpools.
Grundfos would like a model, describing the problem of mixing chemicals,
being dosed into water systems, to be developed. The application of the
model should be dedicated to dosing aqueous solution of chlorine into
The problem is imagined to contain two sub-models. The first model is concerned about dosing a strong aqueous solution of chlorine into a pipe system and the second about injection of purified and chlorinated water into the swimming pool.
The contamination of the water and the chemical process reducing the
chlorine content in the swimming pool could be regarded as uniform and
stationary and dependent of the number of bathers.
Temperature and moisture gradients in sugar silos
The distribution of heat and moisture in sugar stored in silos influence:
Tanaka H, Yoshida K (1984) "Heat and mass transfer mechanisms in a grain storage silo". Engineering Sciences in the food industry. Elsevier Applied Sience Publishers, Essex, England. pp 89-98
Khankari KK, Morey RV, Patankar SV (1994). "Mathematical model for moisture diffusion in stored grain due to temperature gradients". Transactions of the ASAE 37: 1591-1604)